**Definition and Illustration**

The most familiar example of a ring is the set of all integers, **Z**, consisting of the numbers

- . . ., −4, −3, −2, −1, 0, 1, 2, 3, 4, . . .

It serves as a prototype for the axioms for rings. A **ring** is a set *R* equipped with two binary operations + : *R* × *R* → *R* and **·** : *R* × *R* → *R* (where × denotes the Cartesian product), called *addition* and *multiplication*, such that:

- (
*R*, +) is an abelian group with identity element 0, meaning that for all*a*,*b*,*c*in*R*, the following axioms hold:- (
*a*+*b*) +*c*=*a*+ (*b*+*c*) (+ is associative) - 0 +
*a*= a (0 is the identity) *a*+*b*=*b*+*a*(+ is commutative)- for each
*a*in*R*there exists −*a*in*R*such that*a*+ (−*a*) = (−*a*) +*a*= 0 (−*a*is the inverse element of*a*)

- (
- (
*R*,**·**) satisfies- (
*a*⋅*b*) ⋅*c*=*a*⋅ (*b*⋅*c*) (**⋅**is associative)

- (
- Multiplication distributes over addition:
*a*⋅ (*b*+*c*) = (*a*⋅*b*) + (*a*⋅*c*)- (
*a*+*b*) ⋅*c*= (*a*⋅*c*) + (*b*⋅*c*).

As with groups the symbol ⋅ is usually omitted and multiplication is just denoted by juxtaposition.

Although ring addition is commutative, so that *a* + *b* = *b* + *a*, ring multiplication is not required to be commutative; *a* ⋅ *b* need not equal *b* ⋅ *a*. Rings that also satisfy commutativity for multiplication (such as the ring of integers) are called **commutative rings**.

Some basic properties of a ring follow immediately from the axioms.

- The additive identity and the the additive inverse are unique.
- The binomial formula holds for any commuting elements (i.e., ).

Authors do not agree on whether a ring has the multiplicative identity 1 or not; i.e., an element such that . For example, the set of even integers satisfies above axioms and thus constitutes a ring but does not have 1. Rings which *do have* multiplicative identities are sometimes for emphasis referred to as **unital rings**, **unitary rings**, **rings with unity**, **rings with identity** or **rings with 1**. The present article does not go in depth on this issue. The interested readers are referred to the article pseudo-rings.

Following the common practices in Wikipedia, this article adopts the following convention: a commutative ring is assumed to have the identity, while a ring in general is not assumed so. To discuss a commutative ring that is not necessarily unital, we will use the term commutative **Z**-algebra.

Read more about this topic: Ring (mathematics)

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